# Mathematical Finance

## Mathematical Finance

The term “mathematical finance” refers to a set of methods or formulas that can be used to model, quantify and understand phenomena that govern financial operations having a specific time period, such as loans, investments and return calculations, particularly in the area of financial markets.

The variables involved in these phenomena are the amounts invested or borrowed, the duration of the investment or amortization and the simple or compound interest rate applied.

### Formulas

The following variables are used in the points given below:

−  $$C_0$$ : initial value or actual value
−  $$C_n$$ : final value after n periods, or future value
−  $$i_N$$ : nominal or annual interest rate
−  $$i$$ : periodic interest rate expressed in decimal notation
−  $$n$$ : number of periods being considered
−  $$Pmt$$ : value of the annuity (reimbursement or investment)
−  $$S_n$$ : sum of the capitalized values after $$n$$ payments
−  $$S_0$$ : initial value to invest to obtain an annuity of $$Pmt$$

Specific terminology is written in italics.

• Capitalization after 1 year: $$C_n = C_0 + i × C_0$$
• Annual simple interest: $$I_A = C_0 × i$$
• Profit :  $$I_{total} = C_n − C_0$$
• Final value (simple interest): $$C_n = C_0 + (C_0 · i · n)$$
• Return rate (expressed as a percentage): $$Return = \dfrac{Profit}{C_0} · 100$$
• Periodic interest rate : $$i = \dfrac{i_N}{n}$$

Formulas for an initial fixed amount and compound interest

• Initial value: $$C_0 = \dfrac{C_n}{(1 + i)^n}$$
• Final value: $$C_n = C_0 · (1 + i)^n$$
• Number of periods: $$n = \dfrac{\textrm{ln} (\frac{C_n}{C_0})}{\textrm{ln}(1 + i)}$$
• Interest rate: $$i = \sqrt[n]{\dfrac{C_n}{C_0}} \space – 1$$

Formulas for a series of regular payments (Pmt) having compound interest that is reinvested at the end of each period.

• Initial value: $$S_0 = Pmt · \left( \dfrac{1 \space – (1+i)^{-n}}{i} \right)$$
• Final value:  $$S_n = Pmt · \left( \dfrac{(1 + i)^n \space – 1}{i} \right)$$
• Number of periods when the present value is known:: $$n = \dfrac{- \textrm{ln} \left( 1 \space – \frac{S_0}{Pmt} · i \right)}{\textrm{ln} (1 + i)}$$
• Number of periods when the final value is known: $$n = \dfrac{ \textrm{ln} \left( 1 \space + \frac{S_0}{Pmt} · i \right)}{\textrm{ln} (1 + i)}$$
• Amount of the payments when the present value is known formula used to calculate regular repayments $$Pmt$$ of a debt $$C_0$$ :
$$Pmt = S_0 · \left( \dfrac{i}{1 \space – (1 + i)^{-n}} \right)$$
• Amount of the payments when the final value is known (formula used to calculate the regular deposits $$Pmt$$ that allow funds to be generated for a given term $$C_n$$ :
$$Pmt = S_n · \left( \dfrac{i}{(1 + i)^n \space -1} \right)$$